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%I #16 Feb 16 2017 02:44:09
%S 0,0,0,0,1,3,6,13,22,38,58,93,134,202,282,405,554,774,1035,1412,1862,
%T 2489,3243,4267,5496,7137,9106,11684,14782,18782,23575,29689,37010,
%U 46238,57275,71048,87489,107844,132083,161853,197243,240418,291619,353702,427167
%N Difference between sum of largest parts and sum of smallest parts of all partitions of n into an odd number of parts.
%H Alois P. Heinz, <a href="/A211870/b211870.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = A222047(n) - A222044(n).
%F a(n) = A116686(n) - A211881(n).
%e a(6) = 6: partitions of 6 into an odd number of parts are [2,1,1,1,1], [2,2,2], [3,2,1], [4,1,1], [6], difference between sum of largest parts and sum of smallest parts is (2+2+3+4+6) - (1+2+1+1+6) = 17 - 11 = 6.
%p g:= proc(n, i) option remember; [`if`(n=i, n, 0), 0]+
%p `if`(i>n, [0, 0], g(n, i+1)+(l-> [l[2], l[1]])(g(n-i, i)))
%p end:
%p b:= proc(n, i) option remember;
%p [`if`(n=i, n, 0), 0]+`if`(i<1, [0, 0], b(n, i-1)+
%p `if`(n<i, [0, 0], (l-> [l[2], l[1]])(b(n-i, i))))
%p end:
%p a:= n-> g(n, 1)[1] -b(n, n)[1]:
%p seq(a(n), n=0..50);
%t g[n_, i_] := g[n, i] = {If[n==i, n, 0], 0} + If[i>n, {0, 0}, g[n, i+1] + Reverse[g[n-i, i]]]; b[n_, i_] := b[n, i] = {If[n==i, n, 0], 0} + If[i<1, {0, 0}, b[n, i-1] + If[n<i, {0, 0}, Reverse[b[n-i, i]]]]; a[n_] := g[n, 1][[1]] - b[n, n][[1]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 16 2017, translated from Maple *)
%Y Cf. A116686, A211881, A222044, A222047.
%K nonn
%O 0,6
%A _Alois P. Heinz_, Feb 12 2013
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